Skip to main content

Full text of "Analytical Mechanics"

See other formats


306                         ANALYTICAL MECHANICS
(6)   xi = ai sin ~ i and x2 = a2 cos ^j~-(t + to).
(7)   xi = ai cos wi   and   #2 = flfcsin (coZ + 5). ",   (8)  xi = ai cos coi   and   x2 = aa cos (coi + 5).
;    (9) £1 = ai cos coZ   and   #2 = a2 cos ~ (i + Jo).
. (10)  rci = ai sin ( coi + ~ )     and x« = a2 sin —" t.
(11)  a?i = ai sin (coi + 50     and #2 = &2 cos (coi + 52). ""(12y rci = ai cos (coi + 50      and x2 = a2 cos (wi + 52).
^13) xi = ax cos •—•(* + *0) and x2 = a2 sin ~ (I - tQ). t!4) a;i = ai sin ~-(t — tQ) and x2 = a2 sin ^ (t + 1Q). (15) rci = ai cos -—-(I — tQ) and x2 = «2 cos -~~ (i + 150).
240. Elliptic Harmonic Motion. — Consider the motion of a particle which is acted upon by two harmonic forces whose directions are perpendicular to each other. Suppose the periods of vibration of the particle due to the separate action of the forces to be the same, then the following equations define the component motions.
x = a sin o>£,*                                   (1)
y= 6sin(coZ + <5).                            (2)
The equation of the path of the particle may be obtained by eliminating t between equations (1) and (2). Expanding the right-hand member of equation (2) and substituting for sin co£ and cos ut from equation (1) we get
y = b sin ut • cos 5 + b cos ut • sin 5
x                   I      x*
= b - cos 5 + b V 1 -- 1 • sin 5, a                v       a2
* The phase angle is left out of equation (1) to simplify the problem. This, however, does not affect the generality of the problem. It simply amounts to choosing a particular instant as the origin of the time axis. If, however, the phase angle is left out of both of the component motions the generality of the problem is affected because that will amount to assuming that the component motions are in the same phase.